Portfolio Choice and Life Insurance

Transcription

1 Porfolio Choice and Life Insurance H. Huang, M. Milevsky and J. Wang York Universiy 1, 4700 Keele S., Torono, Canada 18 Sepember Huang is an Associae Professor in he Deparmen of Mahemaics and Saisics, Milevsky is he Execuive Direcor of The IFID Cener and Associae Professor in he Schulich School of Business and Wang is a docoral candidae in mahemaics. The conac auhor (M. Milevsky) can be reached a milevsky@yorku.ca,

2 Absrac Porfolio Choice and Life Insurance We sudy a class of porfolio choice problems ha combine life insurance and labor income, wihin he opimal conrol framework pioneered by Meron (1969, 1971). Our model differs from previous research by (i) focusing aenion on he correlaion beween human capial and financial capial, and (ii) modeling he uiliy of he family as opposed o separaing consumpion and beques. From a echnical poin of view we show how he underlying Hamilon-Jacobi-Bellman (HJB) equaion can be simplified using a similariy reducion echnique, which hen allows for he implemenaion of an efficien numerical soluion. And, for reasonable financial economic parameer values, a closed-form approximaion is derived which grealy simplifies he numerical calculaions. A variey of example illusraing our numerical algorihm are also provided. Our main qualiaive resul is ha households whose primary breadwinner s wages are negaively correlaed wih financial marke reurns, should opimally purchase more life insurance and can afford o ake more risky posiions wih heir financial porfolio. In addiion, we find ha he opimal face value of life insurance is remarkably insensiive o he family s risk aversion. KEYWORDS: Acuarial Finance, Personal Risk Managemen, Porfolio Theory, Hamilon- Jacobi-Bellman Equaion, Similariy Reducion, Finie Difference Mehod. JEL Codes: D91, G11

3 1 Inroducion and Moivaion There is a glaring disconnec beween he way mos financial advisers sell life insurance versus how hey sell or promoe invesmen producs such as muual funds, socks and bonds. Aside from he regulaory environmen and he differen licenses required namely securiies as opposed o insurance hese wo financial decisions are presened as if here were a separaion heorem ha jusified heir relaive invariance. Moreover, while he conceps of risk olerance, risk aversion and uiliy are ubiquious in he lingo of he securiies indusry, he same lexicon rarely eners he dialogue in he life insurance arena. This is quie odd, since hisorically he economics of insurance was he breeding ground for much of he developmen in uiliy heory. This observaion is more han jus anecdoal. The insurance lieraure s saring poin for he opimal quaniy of life insurance is he original work by Dr. Solomon Huebner (U. of Penn.) in he early 1900s, based on he concep of a Human Life Value (HLV). This idea is also a he hear of much forensic economics and liigaion, where he cours mus deermine he value of a los life for he purposes of compensaion. See Ziez (2003) and Todd (2004) for a sae of he ar review of he lieraure regarding life insurance. Mos financial planning and invesmen exbooks advocae somehing called a needs analysis where he family deermines how much would be required o mee long-erm expenses and oher financial goals if he primary source of income were los. Oher auhors focus on he presen value of los wages. Regardless of he precise mechanics, rarely is his decision presened as a porfolio choice problem akin o he invesmen in socks and bonds. Our main poin is ha, we believe i should. Afer all, life insurance is a hedge agains he loss of human capial and porfolio hedging decisions should be made joinly, no independenly. Thus, our objecive in his paper is o joinly analyze he decision of how much life insurance a family uni should have o proec agains he loss of i s breadwinner as well as how he family should allocae i s financial resources beween risk-free and risky asses, vis a vis he dynamics of labor income and human capial. We view his problem wihin he paradigm of porfolio choice and use he ools of financial economics pioneered by Meron (1969, 1971) o arrive a opimal conrols for invesmen, consumpion and life insurance bu where he family uni is placed fron and cener. A number of recen papers have exended he se of decisions included in he porfolio choice problem o highligh he ineracion and he risks faced by he household, broadly defined. Thus, for example, Goezmann (1993), Yao and Zhang (2005) as well as Cocco (2005) focus on he role of he housing porfolio; Campbell and Cocco (2003) focus on opimal morgage choices; Sundaresan and Zapaero (1997) examine he role of defined 1

4 benefi pensions, while Dybvig and Liu (2004) model he impac of flexible reiremen daes; Jagannahan and Kocherlakoa (1996) and Viceira (2001) sress he impac of aging; Faig and Shum (2002) are moivaed by he demand for illiquid asses; Koo (1998) as well as Davis and Willen (2000) model he role of labor income; Dammon, Spa and Zhang (2001) focus heir aenion on capial gains and income axes; Heaon and Lucas (2000) focus on he role of enrepreneurial risk. Ohers go back o basics and exend porfolio choice models o include more sophisicaed (and realisic) processes for invesmen reurns, such as Chacko and Viceira (2005) or ime-varying and mean-revering risk premiums, such as Kim and Omberg (1996). And, while he above lis is clearly no exhausive, he unifying heme of he burgeoning porfolio choice lieraure is ha a singular personal financial feaure is highlighed and carefully modeled in order o ease-ou he relevan financial economic insighs relaed o ha one feaure. Some of hese papers go-on o es or calibrae heir models agains real world daa, while ohers end heir conribuion wih a normaive model. All of hem use he Meron (1969, 1971) framework as a saring poin. Our objecive is similar. We sar wih a radiional diffusion model of asse dynamics and uiliy of consumpion, bu we are focused in our modeling of labor income and life insurance purchases. A cenral feaure of our model is he correlaion beween innovaions o he labor income process and financial reurns. We are specifically ineresed in he ineracion beween he uiliy-maximizing demand for life insurance vis he opimal consumpion paern and asse allocaion when here is a non-zero correlaion beween hese wo criical sae variables. We deviae somewha from previous models by focusing on he family uni ha derives uiliy from consumpion and he risk o his family of losing he wage/income source. The emphasis on he family as a uni, which survives independenly of he life saus of he breadwinner, alleviaes he need for specifying a separae uiliy of beques. Tha is no o say ha life insurance has never been analyzed in a financial conex. Early work by economiss such as Yaari (1965), Hakanson (1969), Fischer (1973), Campbell (1980), Lewis (1989), Hurd (1989) and Babbel and Ohsuka (1989) helped he lieraure develop a number of insighs ino he demand for life insurance. These papers illusraed how he naure of moraliy uncerainy, he uiliy and srengh of beques, risk aversion, increasing household wealh and iner-emporal raes of subsiuion impac he decision o purchase life insurance and he ype of life insurance o buy. Bu hese models are no couched in he curren language of porfolio choice no do hey give us much insigh ino he role of labor income dynamics and human capial over he life-cycle. The same is rue of he work by Richards (1975), which exended Meron (1969, 1971) o life insurance, bu wih lile numerical insigh or analyic soluions given he complexiy of he problem. Along he 2

5 same lines, Buser and Smih (1983) adop a porfolio approach o life insurance, bu from a sricly one-period framework. Neverheless, we will reurn o our conribuions above-andbeyond previously known resuls laer on in he analysis, when we presen he specifics of our model. Likewise, our work differs from he recen paper by Chen, Ibboson, Milevsky and Zhu (2005) where he analysis is done in discree ime and via simulaions. One of he main differences beween he curren work and mos previous research is ha we avoid specifying a uiliy of beques ye are sill able o alk ineligenly abou he demand for life insurance. Neveheless, our resuls confirm many of he earlier insighs. From a echnical poin of view, his wo sae variable class of problems wage income and asse prices normally leads o a highly nonlinear Hamilon-Jacobi-Bellman (HJB) parial differenial equaion. Ye, one of conribuions of our paper is ha despie his nonlineariy, analyical soluions can be obained in special cases when wage income and asse prices are perfecly (posiively or negaively) correlaed. When his correlaion is beween -1 and +1, we obain approximae soluions. We firs use he mehod of similariy reducion o reduce he dimension of he HJB equaion. For some special cases we show ha his leads o a se of ordinary differenial equaions, which can be solved explicily. For he general case, we show ha using similariy reducion can eiher help o find approximae soluions or simplify he problem significanly before applying numerical mehods. As a resul, he compuaional cos will be reduced, compared o ha of a full numerical approach applied o he original HJB. Technicaliies aside, our main qualiaive resuls are as follows. Firs, we find he opimal amoun of (face value) life insurance a family should have, depends quie srongly on boh volailiy of he wage/income process as well as i s correlaion o invesmen reurns. Higher volailiy and higher correlaions lead o lower, opimal, life insurance levels. Ceeris paribus, a enured universiy professor needs more life insurance han a Wall Sree invesmen banker, even if hey boh expec o earn he same wages over ime. Furhermore, as one would expec, he risk profile of he wage/income process has a srong impac on he opimal mix beween risky and risk-free asses for he family. This resul should come as no surprise. The original work by Bodie, Meron and Samuelson (1991) all he way o he recen work by Cocco, Gomez and Maenhou (2005) have sressed he financial characerisics of human capial and how i serves as a subsiue for financial capial. Our model easily reproduces his resul and emphasizes he role of life insurance, which is a hedge for human capial. Finally, our model confirms he inuiion shared by many financial planners ha he opimal amoun of life insurance a family should hold is no sensiive o he risk aversion or risk olerance of he family. In our consan relaive risk aversion (CRRA = γ) model, he opimal demand for life insurance is insensiive o γ, even hough he family s financial porfolio is obviously 3

6 quie sensiive o his parameer. As such, our model opens up he possibiliy of esing wheher he acual holding of life insurance is, in fac, correlaed wih oher proxies for a family s risk aversion. Our model provides jusificaion for finding no such relaionship. The res of he paper is organized as follows. The seup of he problem is described in Secion 2 followed by he seup of he HJB equaion in Secion 3. Soluion mehodologies are discussed in Secion 4. In Secion 4.1, we focus on special cases when he wage and risky asse are perfecly correlaed. The general case is discussed in Secion 4.2. We furher simplify he HJB equaion using he similariy reducion echnique, which is followed by he presenaion of he numerical mehod for solving he reduced HJB equaions. Issues of finding he proper boundary condiions are also discussed. In Secion 4.3, we presen a closed form approximae soluion for pracically relevan parameer values. In Secion 5 ypical resuls and he financial implicaions of he resuls are given. We finish he paper wih a discussion of fuure research direcions in Secion 6. The derivaions of he HJB equaion and he closed form soluions are given in he appendices. 2 Model of he Family Life-cycle 2.1 The Inpu Variables The variable denoes he curren ime and we will work wih hree daes of ineres. The firs dae is he ime horizon T H of he family, which is assumed exogenous and deerminisic on he order of 50 o 100 years. The second dae is he ime of reiremen T R < T H, which is when he wage/income process (job) erminaes and he breadwinner eners his or her reiremen years. The hird dae of ineres is he deah of he breadwinner, and end of he wage/income process, which akes place a a random sopping ime denoed by τ. The wage/income process jumps o zero a he minimum of T R and τ. We do no allow flexible reiremen as in Sundaresan (1997). We assume he family purchases shor-erm insurance on he life of he breadwinner which is renegoiaed and guaraneed renewable on an ongoing basis a a pre-deermined schedule which is driven by an insananeous force of moraliy (IFM) curve denoed by λ y+, where y is he age of he primary breadwinner a incepion of he model. We hen le I denoe he insurance premium (in dollars) payable per uni ime; a variable which is under he direc conrol of he family. One can hink of I as a budge for insurance which will hen induce a cerain face value or deah benefi I /λ y+. For example, if he IFM curve a age y = 35 is λ 35 = 0.001, and he family spends I = 50 dollars on life insurance, his eniles he family o a deah benefi of 50/0.001 = $50,

7 We le M denoe he marke value of he family s asses which includes he value of all (risky) socks and (risk-free) bonds on a mark-o-marke basis. We assume ha M 0 denoes he iniial markeable wealh a ime = 0. Heurisically, he breadwinner works and convers labor and ime ino wages and income. A porion of his income is consumed and he remainder is saved in a diversified porfolio consising of risky socks and safe bonds. The variable α denoes he fracional allocaion of he family s markeable wealh M o he risky asse a ime. Thus, if α = 0 he family allocaes all i s markeable wealh bu no more o risk-free bonds and if α = 1 he family allocaes all i s markeable wealh bu no more o risky socks. The model also allows for α > 1 which would imply leverage. Thus, for example, α = 2 implies ha 200% of wealh is invesed in he risky sock. This is financed by borrowing 100% of wealh a some (consan) rae of ineres denoed by r. Recall ha α is under direc conrol of he family and is one of he hree choice variables in our model. We do no address liquidiy consrains and he difficuly faced by younger families when borrowing. Noe ha many of our numerical resuls involve large leverage. We le c denoe he insananeousness consumpion rae of he family (in real erms) per uni ime. In general, our model is specified in real (afer inflaion) erms and all parameers and choice variables will reflec his. Noe ha he consumpion rae is our hird and final choice (a.k.a. conrol) variable and c is chosen o maximize he family s uiliy of consumpion. Even hough a more realisic sae dependen insananeous uiliy funcion can be used, under our PDE-based mehodology, we adoped a simpler CRRA uiliy funcion in his sudy. The precise funcional form is u(c) = 1 1 γ c1 γ (1) for some posiive consan γ > 0, which is labeled he coefficien of relaive risk aversion. Le X denoe he marke value of he risky asse (sock index, marke porfolio) a ime. This sochasic process will be modeled as a geomeric Brownian moion so ha: dx = µ m X d + σ m X db m, (2) where µ m denoes he drif and σ m denoes he diffusion coefficien of he process. This, of course, implies ha ln(x /X 0 ) is normally disribued wih a mean value of (µ m 0.5σ 2 m) and a sandard deviaion of σ. Saed differenly, he geomeric mean reurn (a.k.a. growh rae of he risky asse is µ m 0.5σ 2 m per annum. Typical values of he parameers µ m fall in he range of (5%, 15%) and ypical values of σ m fall in he range of (5%, 50%). The risk-free rae r, which is also he rae a which he family can borrow money, is on he order of magniude of (1%, 3%), which mus be lower han µ m for economic equilibrium purposes. 5

8 We now le W denoe he real (afer-inflaion) wage/income rae of he family s breadwinner per uni ime. This sochasic process is expeced o increase in real erms over ime and migh be correlaed wih he invesmen performance of he risky asse. assume he wage process saisfies he following geomeric Brownian moion (GBM), bu wih specificaion: dw = { We µ w W d + σ w W db W < τ 0 τ, (3) where τ is he random ime of deah, µ w denoes he drif and σ w denoes he diffusion coefficien of he process, and B W denoes he Brownian moion driving he wages process W. Similar o he risky asse, we assume ha he real wage a any fuure ime +s is lognormally disribued wih parameers (µ w, σ w ). We could jus as easily specify a mean-revering process. Noe ha he Brownian moion B m driving he risky asse is insananeously correlaed wih he B W driving he wage process via he relaionship d B m, B W = ρσm σ w d. Laer we will alk abou his correlaion variable ρ, which is he primary focus of our numerical case sudy and resuls. As menioned earlier, he insananeous force of moraliy (or hazard rae) is denoed by λ y+, where y is he iniial age. The quaniy (λ y+ )d can be hough of as he rae of deah wihin a small ime inerval d a ime. The condiional probabiliy of survival, from age y o age y +, under he law of moraliy defined by λ y+ can be compued via: ( p y ) = e 0 (λ y+s)ds, (4) where he noaion on he lef-hand side of he equaion is sandard in he acuarial lieraure. For example, if he fuure lifeime random variable is exponenially disribued, we have ha Pr[τ s] = e λs, and he hazard rae is consan a all ages, a a value of λ y+ = λ. In he general case, he funcion λ y+ is expeced o increases wih ime (age). I is imporan o sress ha in our model he family uni knows exacly how much hey will have o pay for insurance regardless of how much and when hey wan o purchase i from he curren ime, o he ime of reiremen T R. Thus, in addiion o precluding whole-life and oher more complicaed forms of insurance, we do no allow for sochasic moraliy raes or aniselecion effecs which migh complicae he insurance purchase problem. Thus, once again, if he family purchase (invess) I dollars in life insurance a ime, hey will be eniled o a deah benefi of I /λ y+ if he breadwinner dies a ime. For now, we ignore loading and commissions which can easily be handled by working wih a loaded hazard rae λ y+ insead of a biological moraliy rae λ y+. 6

9 2.2 The Financial Wealh Dynamics Based on he consrucion of he wage/income process W and he evoluion of he risky asse price X, he family budge consrain for he markeable wealh process M will saisfy he following sochasic differenial equaion: for < min[τ, T R ], and for > min[τ, T R ]. dm = W d c d I d + α M (µ m d + σ m db m ) + (1 α )rm d (5) = [(µ m r)m α + rm + W c I ] d + σ m α M db m dm = c d + α M (µ m d + σ m db m ) + (1 α )M rd (6) = [(µ m r)m α + rm c ] d + σ m α M db m The inuiion for he various pieces in equaions (5) and (6) is as follows. Firsly, we add wage income when here is some via W d. Secondly, we subrac discreionary consumpion c d and insurance premiums I d. Finally, we add insananeous invesmen reurns from he allocaion o he risky asse, α M dx /X as well allocaions o he risk-free asse (1 α )M rd. Finally, we have subsiued dx /X from equaion (2) o eliminae any reference o X from here on. A he insan of deah = τ, we mus carefully add he deah benefi by disinguishing beween he value of markeable wealh one insan prior o he arrival of deah and one insan afer his arrival. More precisely: M τ+ = M τ + I τ λ y+τ (7) where I τ /λ y+τ is he deah benefi or he face value of he insurance policy a ime τ and λ y+τ is he insananeous force of moraliy, or hazard rae. Finally, he family-uni or household objecive funcion is defined by: [ TH ] max E e δs u(c s )ds F, (8) {α s,i s,c s} where δ is he subjecive discoun rae and u(c) denoes he insananeous uiliy of consumpion a ime. In words, he family is searching for an asse allocaion sraegy α s, an insurance buying sraegy I s, and consumpion sraegy c s ha maximizes he discouned value of uiliy of consumpion beween ime (now) and he erminal horizon of he family uni. We sress once again ha we have eliminaed he need for a uiliy of beques by reaing he family as one uni ha is dependen on he breadwinner for heir source of 7

10 wages. Thus, in conras o he classical insurance models of Yaari (1965), Fischer (1973) or even Campbell (1980), he primary breadwinner is no forced o decide how much hey love heir family, versus how much hey love hemselves. Raher, he family decides o allocae consumpion across he enire horizon of he family, while proecing he wage/income flow using insurance. This assumpion obviously alleviaes he need o measure he srengh of beques, bu implicily assumes he family derives he exac same level of uiliy wheher he primary breadwinner is alive, or no. Of course all psychic value of life is ignored as well. Ineresingly, we find ha even if we change u(c) o HARA uiliy wih Required Lower Bounds on consumpion, resuls can be obained. 3 The Hamilon-Jacobi-Bellman Equaion Wih he model foundaions and formulaion behind us, we proceed in he usual fashion for solving hese problems. Firs, le J(M, W, ) = denoe he indirec uiliy funcion. [ TH ] max E e δs u(c s )ds F, (9) {α s,i s,c s} Now, assume we can find he opimal conrol ha saisfies equaion (9), hen J saisfies he following Hamilon-Jacobi-Bellman (HJB) equaion (we refer he readers o Appendix A for he deailed derivaion): ( ) λ y+ J = J + max e δ u(c ) c J M c ( ( + max Φ M + I ) ), λ y+ I J M I λ y+ + max [α (µ m r)m J M + 12 α α2 σ 2mM ] 2 J MM + α ρσ W σ m W M J W M + rm J M + W J M + µ W W J W σ2 W W 2 J W W (10) where he symbol Φ saisfies he following HJB equaion, bu wihou wage income and life insurance, ( ) 0 = Φ + max e δ u(c ) c Φ M c + max [α (µ m r)m Φ M + 12 α α2 σ 2mM ] 2 Φ MM + rm Φ M. (11) For consan relaive risk aversion uiliy (CRRA), he analyic form of Φ(M, ; T ) iself can be obained as follows Φ(M, ; T ) = h(; T ) M 1 γ 8 1 γ, (12)

11 where h(; T ) saisfies he following ordinary differenial equaion 1 h 1 γ h + γ 1 γ e δ γ h 1 (µ γ m r) 2 + r + 2γσm 2 = 0. (13) Hereafer he prime symbol is used o denoe he derivaive wih respec o ime. analyical soluion of equaion (13) wih zero beques a ime T can be obained as: The ( ) e h(; T ) = e δ ξ(t ) γ 1, (14) ξ where ξ = δ γ + 1 γ (r + (µ ) m r) 2. γ 2γσm 2 Afer applying he firs order condiions, he HJB equaion for J can be rewrien as λ y+ J = J + γ 1 γ c J M + λ y+γ (M + λ 1 1 γ y+i )J M + λmj M + α (µ m r)mj M (α ) 2 σ 2 mm 2 J MM + α ρσ W σ m W MJ W M + rmj M + W J M + µ W W J W σ2 W W 2 J W W. (15) The erminal condiion is J(M, W, T R ) = Φ(M, T R ; T H ). (16) Finally, he opimal conrol c (consumpion), α (allocaion o risky equiy) and I (amoun spen on life insurance purchases) is given by: 4 Soluion Mehodology c = (e δ J M ) 1 γ, (17) α = (µ m r)j M + ρσ m σ w W J W M, (18) σmmj 2 MM [ ( ) ] 1 I JM γ = M λ y+. (19) h(; T H ) In his secion we will consider a special case ρ = ±1, where a closed form analyical soluion can be found. This is he case where human capial (i.e. he sochasic wage process) is eiher 100% posiively or 100% negaively correlaed wih financial marke reurns. 9

12 4.1 Special Case: ρ = ±1 In his case, as shown in Appendix B, he soluion of he HJB (15) equaion akes he form J = where h is given by (14) and k saisfies h()(m + k()w )1 γ 1 γ (20) k k η = 0, (21) k wih η = µ w r λ y+ β(µ m r) and β = ρσ w /σ m. Using he erminal condiion k(t R ) = 0, k can be solved as k = e T η(s)ds R T R e When λ y+ = λ, i.e. a consan moraliy rae, we have The opimal conrol funcions are where β = ρσ w /σ m and x = W/M. k = 1 η I λ y+ M s T R η(n)dn ds. (22) ( e η(t R ) 1 ). (23) = kx, (24) α = µ m r (1 + kx) βkx, (25) c γσ 2 m δ M = e γ h 1 γ (1 + kx) (26) Remark. Since β = ±σ w /σ m for ρ = ±1, η is affeced by he value of ρ, bu no by γ. This relaionship helps o explain, for example, why risk aversion does no impac he face value of life insurance. 4.2 General Case: 1 < ρ < 1 For he general case of correlaed wage and risky asse, he closed form soluion canno be obained. However, we can use he similariy variable x = W/M wihou he resricion of a simple analyical form Similariy reducion We now show ha he wo dimensional HJB equaion can be reduced o a one dimensional one via he similariy reducion echnique. We sar by leing J = M 1 γ F (x, ). 1 γ 10

13 This leads o a new parial differenial equaion, F + c 0 (1 γ)f + c 1 xf x + c 2 x 2 F xx = 0, (27) where c 0 = c M γ + I 1 γ M c 1 = µ w c M The erminal condiion is now: γ 1 γ + α (µ m r) + x + r 1 2 γ (α σ m ) 2, γ 1 γ I M γ λ y+ 1 γ 1 γ x r α (µ m r) α ρσ m σ w γ + (α σ m ) 2 γ, c 2 = 1 2 (α σ m ρσ w ) σ2 w(1 ρ 2 ), ( ) ( ) c 1 = exp δ F xfx γ, M γ 1 γ [ ( ) ] I = h 1 1 γ F xfx γ 1 λ M 1 γ y+, α = (µm r)[(1 γ)f xfx] ρσmσw[γxfx+x2 F xx] σ 2 m[γ(1 γ)f 2γxF x x 2 F xx]. (28) F (x, T R ) = h(t R, T H ). (29) Noe ha he coefficiens c 0, c 1, c 2 and he erminal condiions are funcions of x only. Therefore, he HJB equaion permis he similariy soluion, which is a funcion of x and Numerical Mehod We use he mehod of lines o solve (27), i.e., we approximae he spaial derivaives by finie differences which resuls in a sysem of ordinary differenial equaions wih respec o ime on each grid poin x j F j + c 0 F j + c 1 + c 1 (F j+1 F j ) + c 1 c 1 2δx 2δx (F j F j 1 ) + c 2 δx (F 2 j+1 + F j 1 2F j ) = 0 (30) where δx is he grid size, c 0, c 1 and c 2 are evaluaed on he grid poins, based on he formula in (28). In addiion, we runcae he semi-infinie domain (x > 0) ino a finie one (0 < x < X). Thus we need an exra boundary condiion a x = X. Noe ha he upwind mehod is used for he firs order derivaive erm in x. The sysem of ordinary differenial equaions is hen solved by an appropriae numerical inegraion echnique. For example, we can use a semi-implici mehod where he coefficiens c 0, c 1 and c 2 are evaluaed a he previous ime level and he spaial derivaives are reaed implicily. 11

14 4.2.3 Boundary condiions We observe ha x = 0 corresponds o W = 0, i.e., no wage income. Therefore, he soluion is of a Meron ype (wih he addiion of insurance), which means ha F is no a funcion of x. In oher words, F saisfies he following ordinary differenial equaion: F + c 0 (1 γ)f = 0. (31) A x = X, he siuaion is more complicaed. Moivaed by he soluion for he special case of ρ = ±1 we posulae ha F (1 + kx) 1 γ, F 1 1 γ 1 + kx. In oher worlds, i behaves as a linear funcion of x, hus his by F 1 1 γ where x N = X is where we runcae he domain. ) (F 1 1 γ 0. We can approximae xx N+1 = 2F 1 1 γ N F 1 1 γ N 1 (32) 4.3 Approximae Soluion Moivaed by he soluion for he special case ρ = ±1, we now seek an approximae soluion for he general case 1 < ρ < 1. To proceed formally, we define a new parameer ( ) 2 ɛ = (1 ρ 2 γσm σ w ) µ m r and consider he case when ɛ 1. Noe ha he special cases considered earlier correspond o ɛ = 0. Based on he procedure oulined in Appendix B, we can wrie our soluion in he form J = J 0 + O(ɛ). For J 0 we have he following explici form J 0 = M 1 γ 1 γ h(1 + kx)1 γ, (33) where h and k are given earlier by (14) and (22). For a consan moraliy rae, hey can be solved explicily as [ e h() = e δ ξ(t H ] ) γ 1, ξ (34) k() = 1 ( e η(t R ) 1 ) η (35) wih ξ and η are defined earlier. Recall ha a consan moraliy rae λ y+ = λ, implies ha fuure lifeime is exponenially disribued. 12

15 5 Numerical Resuls and Discussion Wih he model, derivaion and approximaions behind us, we now move-on o presen some case sudies for a realisic se of economic parameers. And so, he parameer values we have chosen for our resuls are as follows: µ m = 7%, σ m = 20%, r = 2%, µ w = 1%, σ w = 5%, M 0 = 1, 200, W 0 = 50, T R = 10, 20, 30, T H = 40, 50, 60, δ = 2%, y = 35, 45, 55. The financial jusificaion for hese numbers are as follows. We assume ha he family can inves heir financial wealh in a broadly diversified index-fund ha is expeced o earn an inflaion-adjused arihmeic mean of µ m = 7% per annum, wih a volailiy of σ m = 20%. These number are consisen wih oher asse allocaion research in he lieraure, see for example Campbell and Viciera (2002) or Chen, Ibboson, Milevsky and Zhu (2005). The risk free rae of r = 2% afer inflaion is also consisen wih curren condiions in he inflaionadjused bond marke. We furher assume ha wages will grow in real (afer inflaion) erms by approximaely µ w = 1% per annum, wih a sandard deviaion of σ w = 5%. This is consisen wih wage inflaion being higher han price inflaion, bu also allows for shocks o wages, such as unemploymen or business cycle effecs. Finally, we assume he family s discoun rae or subjecive rae of ime preference is δ = r = 2%, which is consisen wih various macro economic models. In erms of oher embedded parameers, we assume ha insurance prices (and moraliy raes) are driven by a Gomperz law of moraliy, which a age 45 sars a: λ 45+ = exp{( )/9.5}/9.5. These numbers are consisen wih survival raes implici in pension-based moraliy ables and he analyic law of moraliy has been used in a variey of acuarial and insurance papers, for example Frees, Carriere and Valdez (1996). Eiher way, our firs order of business is o confirm ha when he parameer values are in he above-menioned range, ha our analyic approximaion for ρ ±1 is valid and accurae. Indeed, we are forunae ha his is he case and Figure #1 provides a visual indicaion of his fac by comparing he opimal insurance purchase I0/M 0, as a funcion of he raio beween wages W 0 and markeable wealh M 0, when he correlaion parameer is ρ = 0.5. Noice he close fi beween he analyic (numerical) resul and he analyic approximaion resul. For compleeness, comparison is also given for ρ = 1 as well as for oher conrol variables c 0/M 0 and α0. Figure #1 Placed Here Indeed, based on he above-menioned parameer values, we can compue he value of ɛ = 0.35(1 ρ 2 ), which is small compared o uniy. This explains he close agreemen beween he numerical and analyical soluions. We noe ha for he non-consan moraliy rae 13

16 λ y+, numerical mehods have o be used o compue he funcion k. However, we find ha compared o he full numerical approach, he compuaional cos for compuing k is negligible. As a side noe, he numerical soluion we display was obained using 100 grid poins on a spaial inerval x = [0, 0.25] and a ime sep size of 10 5 on he inerval of = [0, 20]. Our numerical experimens reveal ha he compuaion becomes unsable a a larger ime sep due o oscillaions in he opimal value of α, which is he holding of risky socks. This severe sabiliy consrain may be caused by he semi-explici naure of our algorihm since we compue he coefficiens c, α and I explicily. An alernaive approach would be o use a fully implici mehod. However, i is no clear how much improvemen can be made since an ieraive procedure needs o be used a each ime sep. Despie he fac ha small ime sep sizes have o be used, our numerical mehod is sill compeiive since we have reduced he dimension of he HJB, compared o he mehod in Purcal (2003) where a wo-dimensional HJB was solved direcly. Having esablished he accuracy of he approximae soluion for he paricular se of parameer values lised earlier, we now urn our aenion o he financial implicaions of he soluion. In paricular, we have compued a number of cases which demonsrae he impac of risk aversion and wage correlaion on he opimal asse allocaion and demand for erm life insurance. Table #1 Placed Here More specifically, Table #1 displays he opimal allocaion o risky (marke) socks and he opimal face amoun (deah benefi) for life insurance, assuming he primary breadwinner has T R = 20 years o reiremen and he family horizon is T H = 50 years. This individual is y = 45 years-old, earns W 0 = 50 housand dollars per year in real (afer inflaion) erms and currenly has M 0 = 200 housand dollars saved. These numbers are fairly realisic when we consider he fac ha a 45 year-old should already have some amoun of savings se aside for reiremen purposes. Noice ha o be consisen wih how insurance is discussed by indusry praciioners, we have chosen o display he acual face value of life insurance I /λ y+, insead of he amoun spen on life insurance I or he raio of wealh insurance I /M. The main qualiaive insighs from our model are as follows. Firs, here is a clear link beween he opimal amoun of life insurance and allocaion o socks as a funcion of he correlaion beween (he shock o) wages and he invesmen reurn on socks. For example, a 45-year old who earns $50 housand per annum and who already has $200 in accumulaed savings will opimally purchase beween $762 - $965 housand of life insurance depending on 14

17 he correlaion beween wages (i.e. human capial) and markes (financial capial). If his individual is working in a job/career whose wages profile is couner-cyclical o he financial marke (ρ = 1), hen he opimal amoun of insurance is $964 housand. On he oher hand, if he wage profile is perfecly correlaed wih financial markes, he opimal amoun of life insurance is a lower $762 housand. The economic inuiion for he impac of wagemarke correlaions on he opimal demand for insurance which is also confirm in he recen paper by Chen, Ibboson, Milevsky and Zhu (2005) is ha when human capial is highly correlaed wih financial capial, he uiliy-adjused value of human capial is much lower and hence he family requires less life insurance. A similar sory applies o he impac of his correlaion on he opimal allocaion o risky socks. When he wage process is highly correlaed wih he reurns from he risky asse, he allocaion o financial risky asse is reduced o counerac he exising (and implici) allocaion o risky asses wihin he wage process. Saed differenly (and quie obviously), if your job is very risky in he sense ha your wages depend on he performance of he sock marke, hen you should no be invesing oo much of your financial wealh in he same sock marke. A number of models developed in he early cied papers confirm his. Noice, however, ha he opimal amoun of face value life insurance, I /λ y+, does no depend on he risk aversion of he family. For example, a zero wage o marke correlaion (ρ = 0), he opimal face value is approximaely $855 housand, which is more han 16 imes he breadwinner s annual wage. Noice ha his number is somewha higher han he ofen-heard rule of humb ha people should have 5-8 imes heir annual income in life insurance. Once again, his number does no depend on he level of risk aversion γ, of he family uni or hazard rae. Some economiss migh find i puzzling ha our model does no predic higher levels of life insurance for families ha are more risk averse. Technically his can be sourced o he CRRA uiliy of consumpion ha is idenical wheher he primary breadwinner is alive or dead. However, we believe his resul is fairly consisen wih praciioner inuiion when i comes o life insurance. Namely, a family should be proeced wih a minimal level of insurance regardless of how risk oleran he family (or breadwinner) consider hemselves. Obviously, risk olerance and risk aversion has a very srong impac on he demand for he risky asse α. Families ha are more risk oleran will allocae more financial and markeable wealh o risky equiies, in highly leveraged amouns, as evidenced by our numerical values of α 1. I is no clear a his poin how liquidiy and borrowing consrains would impac hese resuls. Table #2 provides similar resuls, bu wih he key difference ha he individual is now y = 55 years-old and herefore T R = 10 years from reiremen and T H = 40 years from he erminal horizon of he family uni. We assume all oher parameers are exacly he same 15

18 and simply focus on he impac of aging on he opimal face value (i.e. deah benefi) of life insurance and he opimal asse allocaion. Table #2 Placed Here A casual review of he Table #2 reveals subsanially lower values for boh he opimal asse allocaion proporions α, as well as he face amoun of life insurance I /λ y+. Inuiively, he individual is closer o reiremen, wih only 10 more remaining years of work and wages. In his case, he opimal amoun of life insurance lies beween $430 and $480 housand, depending on he wage-marke correlaion ρ. This is only 8 imes he annual wage of W 0 = $50 housand. Noice how he opimal amoun of life insurance coverage closely racks he expeced discouned value of fuure wages. The breadwinner has 10 more years of labor income and buys 8 imes annual income of life insurance. In Table #1 he breadwinner had 20 more years of labor income and he/she purchased 16 imes annual income. In addiion, noice ha once he family uni is 10 years away from reiremen he amoun of financial wealh invesed in he risky asse is grealy reduced. In fac, when he wage-marke correlaion ρ > 0, he family eliminaes leverage and sops borrowing o inves, as evidenced by α < 1 values. Finally, Table #3 akes us back o age 35, which is T R = 30 years prior o reiremen and T H = 60 prior o he end of he family s planning horizon. One oher change in Table #3 is ha we have reduced he iniial (curren) wealh M 0 = 1 in conras o he M 0 = 4W 0 = 100 of he previous wo ables. This paricular parameer assumpion is moivaed by he low levels of wealh one would expec o observe amongs (young) 35 year-old individuals. Table #3 Placed Here In his hird and final example, he amoun of life insurance purchased by he family falls in he millions of dollars, ranging from $1.03 million o $1.45 million depending on he correlaion variable ρ. Wih 30 years o reiremen he family purchases almos 20 imes annual wages in life insurance. Once again, hese numbers are consisen wih a human capial perspecive ha proecs he family agains he loss of he income source. Anoher imporan insigh from Table #3 is he (abnormally) high amouns allocaed o he risky invesmen asse α. Noice ha regardless of he level of risk aversion, wheher i is a high γ or a low γ, he family places housands of a percen in risky socks. This is financed by borrowing housands of a percen, a.k.a. shoring he risky free bond in our model. Clearly, hese very high number are unrealisic (in pracice) and a he very leas are unfeasible given he various insiuional resricions on borrowing wih very lile collaeral. However, we do no believe his (odd) resul represens a flaw or problem wih our model, 16

19 since his ends o plague mos porfolio choice and asse allocaion models in which risk aversion is low, borrowing is unlimied and horizons are long. We refer he ineresed reader o Campbell and Viciera (2002) for an in-deph discussion of he impac of ime horizon, liquidiy consrains on he demand for risky asses, since his falls ouside he main scope of our analysis. 6 Conclusion and Highlighs This paper focused on a subse of porfolio choice problems, where he emphasis was placed on he demand for life insurance as a funcion of labor income. Our primary research quesion was o invesigae he ineracion beween he demand for life insurance, he demand for risky asses and he opimal level of consumpion as funcion of one s occupaion. This life insurance problem has been invesigaed by a number of classical papers in he lieraure, including more recen papers by Chen, Ibboson, Milevsky and Zhu (2005) and Purcal (2003). Our underlying model differs from hose in a number of mehodological and concepual ways, which we have explained in he body of he paper. From a echnical poin of view we show ha he underlying Hamilon-Jacobi-Bellman (HJB) equaion can be simplified using he similariy reducion echnique, which hen allows for he implemenaion of an efficien numerical mehod. Furhermore, for realisic financial economic parameer values, a closed-form approximaion is derived which grealy simplifies he numerical calculaions. Numerical ess confirm he robusness and accuracy of he approximae soluion compared o he full numerical soluions. A variey of financial case sudies illusraing our numerical algorihm are also provided wihin he paper. Our main qualiaive and pracical resul is ha households whose primary breadwinner s wages are negaively correlaed wih financial marke reurns, should opimally purchase more life insurance and can afford o ake more risky posiions wih heir financial porfolio. However, we also find ha he opimal face value of life insurance is remarkably insensiive o he family s risk aversion. From a pracical perspecive our model validaes a number of rules-of-humb used by financial planners when making porfolio and invesmen recommendaions for heir wealh managemen cliens. Firs, we find ha younger invesors i.e. people who are farher away from he dae a which heir employmen wage process will hi zero are more able o olerae financial risk and will herefore inves more in he risky asse compared o he riskfree bond. In fac, we find a very high (opimal) leverage raio ha can range in he hundreds of a percen, for families whose risk aversion is low. Second, we find ha households should insure agains he loss of heir primary breadwinner s wages quie independenly of how risk 17

20 oleran or risk averse he family defines iself. This is consisen wih he Human Life Value (HLV) concep originally inroduced by Dr. Solomon Huebner in he early 1900s. According o his approach o financial planning and insurance, here is some universal muliple of wages he a family should insurance agains losing, regardless of wheher his paricular family likes o gamble. On a slighly less inuiive manner, our model lends credence o he idea of classifying human capial as eiher a sock or a bond. Namely, if he breadwinner s human capial is risky and highly correlaed o he reurns on he equiy marke, hen he family should inves less in he same risky asse. Likewise, a family whose human capial is uncorrelaed wih equiy markes can afford o ake more financial risk wih heir porfolio. Saed simply. If you are sock, you should own more bonds. If you are a bond, you should own more sock. This ype of hinking has only recenly sared o resonae wih financial advisers, bu has sared o appear in relaed aricles in he lieraure. Finally, we find ha he demand for life insurance should depend on he riskiness of he human capial wages process and is correlaion wih financial reurns. High anicipaed levels of wage volailiy and/or high levels of correlaion induce a reducion in he demand for life insurance. Fuure research by he auhors will focus on a variey of exensions o he basic model presened in his paper, ha would furher enriched he lieraures undersanding of porfolio choice problems vis a vis he demand for life insurance. High on our lis is exending our objecive funcion from a CRRA uiliy o a HARA uiliy funcion, where he family uni imposes a baseline consrain, or fixed level of consumpion ha mus be mainained under all circumsances. This baseline or minimal income migh depend on wheher he primary breadwinner is alive, which migh implicily induce a ype of beques moive. Anoher exension involves relaxing he deerminisic force of moraliy assumpion, where here is uncerainy regarding he fuure coss or price of insurance. This migh lead o ye anoher jusificaion for he coexisence and opimaliy of various forms of life insurance, such as erm, whole-life and universal policies as originally explored by Babbel and Ohsuka (1989). Finally, we will exend he model o muliple insurance producs and risky asses each wih heir own correlaion o possibly more han one risky wage process where he porfolio choice problem will involve and elemen of inra-occupaional hedging sraegies. 18

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24 A Derivaion of he HJB Equaion We derive he HJB equaion by using he dynamic programming principle as following. Here we assume in he nex ime period h, he survival probabiliy of he individual is ( h p y+ ), he deah probabiliy is ( h q y+ ), ( h p y+ ) = 1 h q y+ and y is he curren age of he individual. Le h be an arbirarily small ime incremen, we have [ +h ] J(M, W, ) ( h p y+ )E e δs u(c s )ds + J(M +h, W +h, + h) + ( h q y+ )E [ +h e δs u(c s )ds + Φ(M +h + I +h λ 1 y++h, + h) ], (A-1) where Φ(, ) is he opimal objecive funcion when here is no wage income and no need for paying insurance premium. By Io s formula, we have where dj(m, W, ) = (J + A J )d + σ m M J M db m + σ W W J W db W, (A-2) A J = (α (µ m r)m + rm + W c I ) J M + µ W W J W α2 σ 2 mm 2 J MM σ2 W W 2 J W W + α ρσ W σ m M W J W M (A-3) Inegraion equaion (A-2), we obain J(M +h, W +h, + h) = J(M, W, ) + Similarly, for Φ we have +h (J s + A J s )ds + +h σ m M s J M db m s + +h σ W W s J W db W s. (A-4) dφ(m, ) = (Φ + B Φ )d + σ m α M Φ M db m, (A-5) where Inegraing equaion (A-5) yields B Φ = ((µ m r)m α + rm c ) J M σ2 mα 2 M 2 J MM. (A-6) Φ(M +h, + h) = Φ(M, ) + +h (Φ s + B Φ s )ds + +h σ m α s M s Φ M db m s. (A-7) 22

25 Combining (A-1), (A-7) and (A-4), we obain J(M, W, )( h q y+ ) ( h p y+ )E + ( h q y+ )E [ +h [ +h +h ] e δs u(c s )ds + (J s + A J s )ds +h ] e δs u(c s )ds + Φ(M, ) + (Φ s + Bs Φ )ds. (A-8) Dividing by h and rearranging equaion (A-8), as h 0,we obain: ( J(M, W, )λ y+ J + A J + Φ M + I ), λ y+ + e δ u(c ). λ y+ (A-9) The equaliy holds when we ake he opimal conrol: ( ( J + A J + Φ M + J(M, W, )λ y+ = max α,i,c I λ y+, Using equaions (A-3) and (A-10), afer rearrangemen, we obain ) ) λ y+ + e δ u(c ). (A-10) ( ) λ y+ J = J + max e δ u(c ) c J M c ( ( + max Φ M + I ) ), λ y+ I J M I λ y+ + max [α (µ m r)mj M + 12 α α2 σ 2mM ] 2 J MM + α ρσ W σ m W MJ W M + rmj M + W J M + µ W W J W σ2 W W 2 J W W. (A-11) Here Φ saisfies he following HJB (wihou wage and insurance) ( ) 0 = Φ + max e δ u(c ) c Φ M c + max [α (µ m r)mφ M + 12 α α2 σ 2mM ] 2 Φ MM + rmφ M. (A-12) In order o simplify he presenaion, we have dropped he subscrip from M and W. B Derivaion of he Closed Form Soluion B.1 ρ = ±1 Moivaed by he closed form soluion for consan wage Meron (1971), we seek he soluion of he HJB (15) in he form J = h()(m + k()w )1 γ. (B-1) 1 γ 23